平行线和横线

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Triangle Proportionality Theorem
::三角交配性定理

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The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. We can extend this theorem to a situation outside of triangles where we have multiple parallel lines cut by transverals.
::三角比例定理指出,如果一条与三角形一面平行的线横交另一侧,那么它就会按比例分割两侧。我们可以将这个定理扩展至三角形以外的一种情况,在那里,我们有多条平行线被横线截断。

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Theorem: If two or more parallel lines are cut by two transversals, then they divide the transversals proportionally.
::理论:如果两个或两个以上的平行线被两个横向线截断,那么它们就会按比例地将横向线分割开来。

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If $\begin{align*}l \parallel m \parallel n\end{align*}$ , then $\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}$ or $\begin{align*}\frac{a}{c} = \frac{b}{d}\end{align*}$ .
::如果 lmn, 那么 ab=cd 或 ac=bd 。

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Note that this theorem works for any number of parallel lines with any number of transversals. When this happens, all corresponding segments of the transversals are proportional.
::请注意, 此定理适用于与任何多个横向线的任何数条平行线。 当发生这种情况时, 横切线的所有相应部分都是比例的 。

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What if you were looking at a map that showed four parallel streets (A, B, C, and D) cut by two avenues, or transversals, (1 and 2)? How could you determine the distance you would have to travel down Avenue 2 to reach Street C from Street B given the distance down Avenue 1 from Street A to Street B, the distance down Avenue 1 from Street B to C, and the distance down Avenue 2 from Street A to B?
::如果你看到一张地图,显示四条平行街道(A、B、C和D)被两条道路截断,或横跨两条道路(1和2)的地图,你如何确定从B街到B街到C街的距离? 鉴于1号大道从A街到B街的距离、1号大道从B街到C街的距离、2号大道从B街到B街的距离,以及2号大道从A街到B街的距离,你如何确定你必须从2号大道到C街的距离?

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Examples
::实例

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Example 1
::例1

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Find $\begin{align*}a, b,\end{align*}$ and $\begin{align*}c\end{align*}$ .
::寻找 a, b, c.

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Line up the segments that are opposite each other.
::排成一行的相对部分。

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$$\begin{align*}\frac{a}{9} &= \frac{2}{3} && \quad \ \frac{2}{3} = \frac{4}{b} && \quad \ \frac{2}{3} = \frac{3}{c}\\ 3a &= 18 && \quad 2b = 12 && \quad 2c = 9\\ a &= 6 && \quad \ \ b = 6 && \quad \ c = 4.5\end{align*}$$
::a9=23 23=4b 23=3c3a=182b=122c=9a=9a=6b=6c=4.5

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Example 2
::例2

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Below is a street map of part of Washington DC. $\begin{align*}R\end{align*}$ Street, $\begin{align*}Q\end{align*}$ Street, and $\begin{align*}O\end{align*}$ Street are parallel and 7 $\begin{align*}^{th}\end{align*}$ Street is perpendicular to all three. All the measurements are given on the map. What are $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ ?
::下面是华盛顿市R街、Q街和O街的街道地图,七街与所有三街平行,所有测量都标在地图上。什么是 x 和 y ?

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To find $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ , you need to set up a proportion using the parallel lines.
::要找到 x 和 y, 您需要使用平行线设置比例 。

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$$\begin{align*}\frac{2640}{x} = \frac{1320}{2380} = \frac{1980}{y}\end{align*}$$
::2640x=1320280=1980y

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From this, $\begin{align*}x = 4760 \ ft\end{align*}$ and $\begin{align*}y = 3570 \ ft\end{align*}$ .
::从这里, x=4760英尺和y=3570英尺。

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Example 3
::例3

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Find $\begin{align*}a\end{align*}$ .
::找一个

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The three lines are marked parallel, so to solve, set up a proportion.
::这三条线是平行的,所以要解决,确定比例。

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$$\begin{align*}\frac{a}{20} &= \frac{9}{15}\\ 180 &= 15a\\ a &= 12\end{align*}$$
::a20=915180=15aa=12

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Example 4
::例4

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Find $\begin{align*}b\end{align*}$ .
::查找b。

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To solve, set up a proportion.
::要解决, 设定比例 。

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$$\begin{align*}\frac{12}{9.6} &= \frac{b}{24}\\ 288 &= 9.6b\\ b &= 30\end{align*}$$
::129.6=b24288=9.6bb=30

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Example 5
::例5

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Find the value of $\begin{align*}x\end{align*}$ that makes the lines parallel.
::查找使线条平行的 x 值 。

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To solve, set up a proportion and solve for $\begin{align*}x\end{align*}$ .
::要解析, 设置比例和解析 x 。

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$$\begin{align*}\frac{5}{8} = \frac{3.75}{2x-4} \longrightarrow \ 5(2x-4) &= 8(3.75)\\ 10x-20 &= 30\\ 10x &= 50\\ x &= 5\end{align*}$$
::58=3.752x--45(2x-4)=8(3.7510x-20=3010x=50x=5)

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Review
::回顾

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Find the value of each variable in the pictures below.
::在下面的图片中查找每个变量的值 。

1. 
2. 
3. 
4. 
5. 

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The street map shows part of New Orleans. Burgundy St., Dauphine St. and Royal St. are parallel to each other. If Spain St. is perpendicular to all three, find the indicated distances.
::街道地图显示了新奥尔良的一部分。 Burgundy街、 Dauphine街和Royal St. 之间是平行的。 如果西班牙街与所有这三家都垂直, 请找到指定的距离 。

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6. What is the distance between points $\begin{align*}A\end{align*}$ and $\begin{align*}B\end{align*}$ ?
   ::A点和B点之间的距离是多少?
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7. What is the distance between points $\begin{align*}C\end{align*}$ and $\begin{align*}D\end{align*}$ ?
   ::C点和D点之间的距离是多少?
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8. What is the distance between points $\begin{align*}A\end{align*}$ and $\begin{align*}D\end{align*}$ ?
   ::A点和D点之间的距离是多少?

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Using the diagram, answer the questions.
::使用图表,回答问题。

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9. What is the value of $\begin{align*}w\end{align*}$ ?
   ::W值是多少?
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10. What is the value of $\begin{align*}x\end{align*}$ ?
    ::x 值是什么 ?
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11. What is the value of $\begin{align*}y\end{align*}$ ?
    ::y的价值是什么?
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12. What is the length of $\begin{align*}\overline{AB}\end{align*}$ ?
    ::AB的长度是多少?
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13. What is the length of $\begin{align*}\overline{AC}\end{align*}$ ?
    ::AC的长度是多少?

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Using the diagram, fill in the blank.
::使用图表,填入空白。

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14. If $\begin{align*}b\end{align*}$ is one-third $\begin{align*}d\end{align*}$ , then $\begin{align*}a\end{align*}$ is ____________________.
    ::如果b为三分之一d,则a为________________________。
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15. If $\begin{align*}c\end{align*}$ is two times $\begin{align*}a\end{align*}$ , then $\begin{align*}b\end{align*}$ is ____________________.
    ::如果c为2乘以1,b为______________________________________________________________________________________

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Review (Answers)
::回顾(答复)

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Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。